Theorem eqmadd2 ≪ | index | src | ≫ 

theorem eqmadd2 (a b c n: nat): $ mod(n): a + b = a + c <-> mod(n): b = c $;
StepHypRefExpression
1 bitr 
(mod(n): a + b = a + c <-> mod(n): b + a = c + a) -> (mod(n): b + a = c + a <-> mod(n): b = c) -> (mod(n): a + b = a + c <-> mod(n): b = c)
2 eqmeq 
n = n -> a + b = b + a -> a + c = c + a -> (mod(n): a + b = a + c <-> mod(n): b + a = c + a)
3 eqid 
n = n
4 2, 3 ax_mp 
a + b = b + a -> a + c = c + a -> (mod(n): a + b = a + c <-> mod(n): b + a = c + a)
5 addcom 
a + b = b + a
6 4, 5 ax_mp 
a + c = c + a -> (mod(n): a + b = a + c <-> mod(n): b + a = c + a)
7 addcom 
a + c = c + a
8 6, 7 ax_mp 
mod(n): a + b = a + c <-> mod(n): b + a = c + a
9 1, 8 ax_mp 
(mod(n): b + a = c + a <-> mod(n): b = c) -> (mod(n): a + b = a + c <-> mod(n): b = c)
10 eqmadd1 
mod(n): b + a = c + a <-> mod(n): b = c
11 9, 10 ax_mp 
mod(n): a + b = a + c <-> mod(n): b = c

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)